Tuesday, June 29, 2010

A Simple Astronomical Linear Visual Magnitude Scale and Conversion

The classification of stars as having one of six magnitudes of brightness, with stars of the first magnitude being the brightest and stars of the sixth magnitude those barely visible to the naked eye, and each increase of one in magnitude associated with a rough halving of brightness, is attributed to Hipparchus (c. 190-120 BC), was developed and popularized by Ptolemy (c. 90-180 AD), and was revised by Pogson (March 23, 1829 – June 23, 1891) to represent stars of the first magnitude as one hundred times brighter than those of the sixth, each decrease of one magnitude implying a 2.51-fold increase in brightness (and each increase in magnitude an inverse 0.398-fold decrease in brightness), 2.51 to the fifth power equalling 100 (and 1/2.51 equalling 0.398).

The advantages of this system are its establishment and consequent familiarity, and the ease with which it accommodates the ever-dimmer objects revealed by ever-better telescopes and imaging techniques by simply adding new and higher magnitudes.

Its disadvantages are its inverted representation of brightness, with lower magnitudes representing greater and higher magnitudes representing lesser brightnesses, and its representation of a non-logarithmic phenomenon, brightness or difference in brightness, as logarithmic.

A relatively simple linear representation of brightness, with higher magnitude indicating greater brightness, and double or half the magnitude indicating double or half the brightness respectively, and so on, and an equally simple formula for conversion of standard magnitudes to that representation, are given here.

The relationship of ordinary integer magnitudes 1-11 to the new can then tabulated as follows, to three significant figures:

vmagvlmag

1 1,000,000

2 398,000

3 158,000

4 63,100

5 25,100

6 10,000

7 3,980

8 1,580

9 631

10 251

11 100

Note that for each increase in ordinary magnitude, the linear magnitude is decreased by a factor of 1/2.51 or 0.398, and, inversely, that for each decrease in ordinary magnitude, the linear magnitude is increased 2.51-fold.

Note too the "cycling" of significant figures as you go up or down the table; can you "fill in the blanks" for ordinary magnitudes 12-15?

The formula for converting ordinary (logarithmic) magnitudes to the new linear magnitude above is

vlmag = 1,000,000 * 2.51 ^ - ( vmag - 1 )

However, since 1,000,000 = 100^3 = (2.51^5)^3 = 2.51^15, this can be simplified greatly:

Note that any magnitude might be set to 1,000,000 above and the entire linear scale shifted in relation to the ordinary, and that that starting magnitude might be set to any other convenient power of 100 (to retain such simplification as above).

Thus the scale given above might be distinguished where need be as the vlmag1 scale, referring to the ordinary first magnitude being set as the initial highest linear magnitude value, or it might be distinguished as the vlmag6 scale, referring to the setting of the ordinary first magnitude to one million (10^6), or even as the vlmag1-6 scale.